Counting fluctuations of Lee–Yang zeros: an exactly solvable model with prime-logarithmic fields

We study the zeros of the partition function of ferromagnetic Ising models in which the i-th spin is coupled to a complex field h_i(s) = (s − 1/2)λ_i with λ_i > 0. The motivating case is λ_i = log p_i, where p_i is the i-th prime. By the Lee–Yang theorem all zeros lie on the line Re s = 1/2, where they form a Fourier quasicrystal. Our main results concern their counting function. Exact identity and bound. The counting error D_N(T), measured from the linear main term, equals 1/π times the argument of the fugacity polynomial along a linear flow on the Lee–Yang torus, and |D_N(T)| ≤ N/2 with an optimal constant. Exact statistics. When the λ_i are linearly independent over the rationals (for primes, by unique factorization), the flow is equidistributed and D_N acquires an exact statistical theory. General ferromagnetic graphs. D_N is a sum of bounded martingale differences, with mean zero and variance at most N/12. The fluctuations of the zero count equal the fluctuations of the free energy on the torus. Open chain. The long-time distribution of D_N is exactly that of a sum of N independent random variables, with closed-form variance and supremum and a central limit theorem. This parallels Selberg's central limit theorem for the argument of the Riemann zeta function, with each prime contributing O(1) rather than O(1/p) to the variance. We also delimit the analogy with ζ: at fixed temperature, no holomorphic limit exists near the critical line; the limit in Re u > 1 is a Dirichlet series over squarefree integers (for uncoupled spins, the fermionic primon gas ζ(u)/ζ(2u)); the functional equation of ζ is centred on Re u = 1/2, not on the Lee–Yang line Re u = 0. Exact transfer-matrix computations confirm the counting identity and the open-chain law, and show a crossover from Poisson to picket-fence statistics without GUE behaviour. This record contains the English and Chinese versions (PDF) and their LaTeX sources.